# Games over Probability Distributions Revisited: New Equilibrium Models and Refinements

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Our Contribution

- The proposal to model utilities not only as numbers, but whole distributions. This was first proposed in [18], and applied in [3,10,19]. The theoretical pathologies arising with this, however, were not discussed in these past studies, until reference [16] first identified theoretical difficulties. This prior reference did not provide solutions for some of the issues raised, which our work does.
- The use disappointment rates in game theory. Prior work did either not consider disappointment in the context of games [20,21], or focused on the applicability of conventional equilibria by means of approximation or using endogenous sharing [22]. Our study differs by describing and proving how to make disappointment rates continuous, and for the first time, proposes disappointment as a (novel) equilibrium selection criterion.
- We show the lexicographic Nash equilibrium as a solution concept that is well defined for vector-valued games that past work has shown to not necessarily have a classical Nash equilibrium [16]. Our work picks up this past example of a game without an equilibrium and shows how to define and compute a meaningful solution.
- We show and explain a case of non-convergence of Fictitious Play (FP) in zero-sum games that has been reported in [23] but was left unexplained ever since in past research. This phenomenon appears, as so-far known, only in games with distributions as utilities, since FP is known to converge for zero-sum games by classical results [24].

#### 1.2. Preliminaries and Notation

#### 1.3. Disappointment Rates

- The game is designed to advise the defender to best protect its system against an attacker. The optimization will thus be a minimization of the expected loss ${E}_{\mathbf{x},\mathbf{y}}\left(L\right)$ for the defender, accomplished at the equilibrium strategy $\mathbf{x}={\mathbf{x}}^{*}$ and best reply to it ${\mathbf{y}}^{*}$ in a security game model. See [25] for a collection of examples.
- Knowing that it has to prepare for an expected loss of $v=E\left(L\right)$, the defender may build up backup resources to cover for cases of a loss $>v$, one way of which is buying insurance for it. However, $E\left(L\right)$ is an average, and naturally, there may be infinitely many incidents where the loss overshoots the optimized expectation v that an insurance may cover. Hence, naturally, the defender will strive to minimize the chances for the current loss being $>v$. This is where disappointment rates come into the game.

**Example**

**1.**

#### 1.4. Making the Disappointment Rate a Continuous Function

**Proposition**

**1.**

## 2. Games with Rewards Expressed as Whole Distributions

**Remark**

**1**

**.**It is common in the literature to denote the hyperreal space by a superscript ∗ preceding $\mathbb{R}$, i.e., to use the symbol ${}^{*}\mathbb{R}$. The superscripted star is, however, also commonly used to denote optimal strategies in game theory, such as ${\mathbf{x}}^{*}$ as the best strategy for some players. To avoid confusion about the rather similar notation here, we therefore write $\widehat{x}$ to mean a hyperreal element, and reserve the ∗-annotation for variables to mark them as “optima” for some minimization or maximization problem, with the symbol ${}^{*}\mathbb{R}$ being the only exception, but without inducing ambiguities.

**Definition**

**1**

**.**Let ${}^{*}\mathbb{R}$ be a space of hyperreals (induced by an arbitrary but fixed ultrafilter). Let $X\sim F,Y\sim G$ be random variables with distributions $F,G$ that have moments of all orders. To both, X and Y, associate the hyperreal number $\widehat{f}={\left(E\left({X}^{k}\right)\right)}_{k\in \mathbb{N}}$ and $\widehat{g}={\left(E\left({Y}^{k}\right)\right)}_{k\in \mathbb{N}}$. The hyperreal stochastic order ${\le}_{hr}$ is defined between $X{\le}_{hr}Y$ as $X{\le}_{hr}Y\stackrel{\mathit{def}}{\iff}\widehat{f}\le \widehat{g}$ in the given instance of ${}^{*}\mathbb{R}$.

- We let the payoffs be distributions that have moments of all orders.
- These distributions are replaced by their hyperreal representatives.
- The reward space is ${}^{*}\mathbb{R}$, with its total ordering ≤ on it.
- By the transfer principle, ${}^{*}\mathbb{R}$ “behaves like $\mathbb{R}$”, and hence the concepts and results from game theory “carry over”.

- $\widehat{\mathbf{x}}=({\widehat{x}}_{1},\dots ,{\widehat{x}}_{n}),\widehat{\mathbf{y}}=({\widehat{y}}_{1},\dots ,{\widehat{y}}_{n})$ are vectors of hyperreals that satisfy the same conditions as all (categorical) probability distributions do, i.e., ${\widehat{x}}_{i}\ge \widehat{0}$ and ${\widehat{x}}_{1}+{\widehat{x}}_{2}+\dots +{\widehat{x}}_{n}=\widehat{1}$ (likewise for the ${\widehat{y}}_{i}$s); only to be taken inside the hyperreal space.
- $\widehat{\mathbf{A}}$ is the matrix of payoffs, again composed from hyperreal numbers.

#### 2.1. Obtaining the Payoff Distributions

#### 2.2. Computing Equilibria Is (Not Automatically) Possible

#### 2.3. The Hyperreal ${\le}_{hr}$ as a Tail Order

**Lemma**

**1**

**.**Let $f:[a,b]\to \mathbb{R}$ be a continuous probability density function supported on the compact interval $[a,b]\subset \mathbb{R}$. Then, for every $\epsilon >0$, there is a piecewise polynomial probability density g that uniformly approximates f as ${\u2225f-g\u2225}_{\infty}<\epsilon $.

- C1:
- Criterion for categorical distributions: let $f=({p}_{1},\dots ,{p}_{n}),g=({q}_{1},\dots ,{q}_{n})$ be categorical distributions on a common and strictly ordered support $\Omega =\left\{1,2,\dots ,n\right\}$. Then, we have$$f{\le}_{hr}g\u27fa({p}_{n},{p}_{n-1},\dots ,{p}_{1}){\le}_{\mathrm{lex}}({q}_{n},{q}_{n-1},\dots ,{q}_{1}),$$
- C2:
- Criterion for continuous distributions ([37], Proposition 4.3): let f satisfy condition (8). For every such f, we can calculate a finite-dimensional vector ${\mathbf{v}}_{f}\in {\mathbb{R}}^{n}$, where n depends on f, with the following property: given two density functions $f,g$ with computed vectors ${\mathbf{v}}_{f}=({v}_{f,1},\dots ,{v}_{f,n})$ and ${\mathbf{v}}_{g}=({v}_{g,1},\dots ,{v}_{g,m})$, we have $f{\le}_{hr}g$ if and only if ${\mathbf{v}}_{f}{\le}_{\mathrm{lex}}{\mathbf{v}}_{g}$, taking absent coordinates to be zero when $n\ne m$. If the two vectors are lexicographically equal, then $f=g$.

**C1**. We will look into this next.

## 3. A Game with Continuous Vector-Valued Payoffs but without a Nash Equilibrium

## 4. Lexicographic Nash Equilibria as a Solution Concept

- Consider several goals ${u}_{1}\succ {u}_{2}\succ \dots \succ {u}_{d}$ in a strict order of priority ≻, with ${u}_{1}$ being the most important goal, and all these goals mapping mixed strategies into $\mathbb{R}$-valued payoffs.
- Provide a strategy such that any attempt to increase the revenue in ${u}_{i}$ will enable the other player to find a strategy ${\mathbf{y}}_{i}^{\prime}$, so that the expected payoffs for player 1 become reduced to ${u}_{j}({\mathbf{x}}^{\prime},{\mathbf{y}}_{i}^{\prime})<{u}_{j}({\mathbf{x}}^{*},{\mathbf{y}}^{*})$ for a more important goal ${u}_{j}\succ {u}_{i}$.

**Proposition**

**2.**

**Proof.**

**Definition**

**2**

**.**Let $\left\{{\mathbf{A}}_{i}\in {\mathbb{R}}^{n\times m}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}i=1,2,\dots ,d\right\}$ be a finite collection of matrices that define payoff functions $\mathbf{u}$ in a two-player zero-sum game, all over the same strategy spaces $A{S}_{1},A{S}_{2}$ for the players, and listed in descending order of priority (i.e., ${\mathbf{A}}_{1}$ is the most important, and ${\mathbf{A}}_{d}$ is the least important goal dimension). We call a strategy profile $({\mathbf{x}}^{*},{\mathbf{y}}^{*})\in \u25b5\left(A{S}_{1}\right)\times \u25b5\left(A{S}_{2}\right)$ a lexicographic Nash equilibrium in mixed strategies, if it is best reply to itself, i.e., a fixed point of the best response correspondence $(\mathbf{x},\mathbf{y})\mapsto B{R}_{{u}_{1:d}}({\mathbf{x}}^{*},{\mathbf{y}}^{*})\times B{R}_{-{u}_{1:d}}({\mathbf{y}}^{*},{\mathbf{x}}^{*})$, with $BR$ defined by (11).

**Remark**

**2.**

#### 4.1. Convergence (Failure) of FP in Zero-Sum Games

**Example**

**2**

**.**Let the expected payoffs in a hypothetical $2\times 2$-game be given as

#### 4.2. Online Learning Lexicographic Nash Equilibria in Zero-Sum Games

**C1**above) or otherwise representative values computable for a continuous distribution (see criterion

**C2**above). So, the payoff matrix for a game with payoffs represented as full probability distributions is a matrix of vectors $\mathbf{A}={\left({\mathbb{R}}^{d}\right)}^{n\times m}$, and in which the payoffs are strictly ordered lexicographically.

#### 4.3. Exact Computation of Lexicographic Nash Equilibria in Two-Player Zero-Sum Games

**Remark**

**3.**

## 5. Summary

- Define the payoff distributions over a joint and compact support $\Omega =[a,b]$ that is a closed interval over the reals, or a common discrete interval $[a,a+1,\dots ,b-1,b]\subset \mathbb{N}$, with $a\ge 1$ in both cases.
- If the payoff distributions are continuous, they should either have piecewise polynomial density functions, or be approximated by such functions, which is possible up to arbitrary precision by Lemma 1.
- The games themselves may not admit Nash equilibria as mixed strategies with real-valued probabilities, but do have lexicographic Nash equilibria that are efficiently computable by a series of linear optimizations (see Section 4.3) using existing software [44], at least for finite zero-sum games. Care has to be taken to not misinterpret the resulting strategy profiles as conventional Nash equilibria. In particular, the solution does not need to be lexicographically optimal (as shown in Section 3), but is a best reply to itself under an accordingly modified reply-correspondence (11).

## 6. Outlook

#### 6.1. Phenomena That May Merit Future Studies

- Convergence failure of FP in the sense that the iteration may converge to an accumulation point that is not an equilibrium, despite the fact that the iteration, when carried out towards hyperreal infinity, would converge to an equilibrium; only it is not reachable from within iterating in $\mathbb{N}$, resp. $\mathbb{R}$. This phenomenon is due to the fact that $\mathbb{N}$ is a strict subset of its hyperreal counterpart ${}^{*}\mathbb{N}$, in which FP does converge (by the transfer principle).
- Games that lack a classical Nash equilibrium, although they do have continuous payoffs, with the only difference that they are vector valued. In the absence of conventional Nash equilibria, lexicographic optimization and equilibria, as proposed in Definition 2, may be a solution concept to consider. Intuitively, we hereby treat multiple goals as equilibrium selection criteria, signifying one goal as the most important, and using the subordinate goals to merely refine the equilibrium strategies if they are ambiguous.

#### 6.2. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FP! | Fictitious Play. |

## Appendix A. Proof of Proposition 1

**Term****(I):**- put $t=u\left(s\right)={E}_{{F}_{s}}\left(L\right)$ and assume ${\u2225{s}^{\prime}-s\u2225}_{\infty}<{\delta}_{1}$. We have ${\sum}_{i,j}{x}_{i}{y}_{j}{F}_{ij}\left(t\right)-{\sum}_{i,j}{x}_{i}^{\prime}{y}_{i}^{\prime}{F}_{ij}\left(t\right)={\sum}_{ij}({x}_{i}{y}_{i}-{x}_{i}^{\prime}{y}_{i}^{\prime}){F}_{ij}\left(t\right)$ and furthermore $|{x}_{i}{y}_{j}-{x}_{i}^{\prime}{y}_{i}^{\prime}|=|{x}_{i}{y}_{j}-{x}_{i}^{\prime}{y}_{j}^{\prime}-{x}_{i}^{\prime}{y}_{j}+{x}_{i}^{\prime}{y}_{j}|\le |{y}_{j}({x}_{i}-{x}_{i}^{\prime})|+|{x}_{i}^{\prime}({y}_{j}-{y}_{j}^{\prime})|\le {\delta}_{1}|{y}_{j}|+{\delta}_{1}\left|{x}_{i}^{\prime}\right|\le 2{\delta}_{1}$, because $|{y}_{j}|,|{x}_{i}^{\prime}|\le 1$ for every categorical distribution (to which ${y}_{j}$ and ${x}_{i}^{\prime}$ are the respective probability masses).Applying this bound on the sum gives$$\begin{array}{ccc}\hfill \left|\sum _{ij}{x}_{i}{y}_{j}{F}_{ij}\left(t\right)-\sum _{ij}{x}_{i}^{\prime}{y}_{i}^{\prime}{F}_{ij}\left(t\right)\right|& =& \left|\sum _{ij}({x}_{i}{y}_{j}-{x}_{i}^{\prime}{y}_{j}^{\prime}){F}_{ij}\left(t\right)\right|\hfill \\ & \le & \sum _{ij}|{x}_{i}{y}_{j}-{x}_{i}^{\prime}{y}_{i}^{\prime}|{F}_{ij}\left(t\right)\hfill \\ & \le & 2{\delta}_{1}\xb7\sum _{ij}{F}_{ij}\left(t\right)\le 2{\delta}_{1}\xb7n\xb7m=:\epsilon ,\hfill \end{array}$$
**Term****(II):**- the mapping ${\mathbf{s}}^{\prime}=({x}^{\prime},{y}^{\prime})\mapsto F\left(t\right)={\sum}_{ij}{x}_{i}^{\prime}{y}_{j}^{\prime}{F}_{ij}\left(t\right)$ delivers a function, which is uniformly continuous as a function of t on its entire support, since:
- All ${F}_{ij}$ were assumed to be continuous and compactly supported,
- So they are all uniformly continuous, and $\epsilon >0$ admits individual values ${\delta}_{ij}>0$, so that $|t-{t}^{\prime}|<{\delta}_{ij}$ implies $|F\left(t\right)-F\left({t}^{\prime}\right)|<\epsilon $ for all $t,{t}^{\prime}$ within the support.
- Since the game is finite, we can put ${\delta}_{2}:=min\left\{{\delta}_{ij}\right\}$ to conclude that ${\u2225t-{t}^{\prime}\u2225}_{\infty}<{\delta}_{2}$ further implies $|{F}_{{s}^{\prime}}\left(t\right)-{F}_{{s}^{\prime}}\left({t}^{\prime}\right)|<\epsilon $, i.e., the function F is, independently of ${s}^{\prime}$ continuos with the same $\epsilon ,{\delta}_{2}$, since: ${F}_{s}\left(t\right)={\sum}_{ij}{x}_{i}{y}_{j}{F}_{ij}\left(t\right)$ and so $|{F}_{s}\left(t\right)-{F}_{{s}^{\prime}}\left(t\right)|\le {\sum}_{ij}{x}_{i}^{\prime}{y}_{j}^{\prime}\left|{F}_{ij}\left(t\right)-{F}_{ij}\left({t}^{\prime}\right)\right|<{\sum}_{ij}{x}_{i}^{\prime}{y}_{i}^{\prime}\xb7\epsilon =\epsilon \xb7{\sum}_{ij}{x}_{i}^{\prime}{y}_{j}^{\prime}=\epsilon $, as long as $|t-{t}^{\prime}|<{\delta}_{2}$.

Hence, the family $\left\{{F}_{s}\left(t\right):s\in \Delta \left(A{S}_{1}\right)\times \Delta \left(A{S}_{2}\right)\right\}$ is in fact equicontinous in t (over its support).

## Appendix B. Proof of Lemma 1

## Appendix C. Fictitious Play Algorithm

**C1**and

**C2**from Section 2.3. It is an adapted version from [23].

Algorithm A1 Fictitious Play | ||

Require: an $(n\times m)$-matrix $\mathbf{A}$ of payoff distributions $\mathbf{A}=\left({F}_{ij}\right)$ | ||

Ensure: an approximation $(\mathbf{x},\mathbf{y})$ of an equilibrium pair $({\mathbf{p}}^{*},{\mathbf{q}}^{*})$ and two distributions ${v}_{low},{v}_{up}$ so that ${v}_{low}{\le}_{hr}F({\mathbf{p}}^{*},{\mathbf{q}}^{*}){\le}_{hr}{v}_{up}$. Here, $F({\mathbf{p}}^{*},{q}^{*})\left(r\right)=Pr(R\le r)={\sum}_{i,j}{F}_{ij}\left(r\right)\xb7{p}_{i}^{*}{q}_{j}^{*}$. | ||

1: initialize $\mathbf{x}\leftarrow \mathbf{0}\in {\mathbb{R}}^{n}$, and $\mathbf{y}\leftarrow \mathbf{0}\in {\mathbb{R}}^{m}$ | ||

2: ${v}_{low}\leftarrow $ the minimum over all column-maxima | ||

3: $r\leftarrow $ the row index giving ${v}_{low}$ | ||

4: ${v}_{up}\leftarrow $ the maximum over all row-minima | ||

5: $c\leftarrow $ the column index giving ${v}_{up}$ | ||

6: $\mathbf{u}\leftarrow ({F}_{1,c},\dots ,{F}_{n,c})$ | ||

7: ${y}_{c}\leftarrow {y}_{c}+1$ | ▹ $\mathbf{y}=({y}_{1},\dots ,{y}_{m})$ | |

8: $\mathbf{v}\leftarrow \mathbf{0}$ | ▹ initialize $\mathbf{v}$ with m functions that are zero everywhere | |

9: for $k=1,2,\dots $ do | ||

10: ${u}^{*}\leftarrow $ the minimum of $\mathbf{u}$ | ||

11: $r\leftarrow $ the index of ${u}^{*}$ in $\mathbf{u}$ | ||

12: ${v}_{up}\leftarrow $ the maximum of $\left\{{\mathbf{u}}^{*}/k,{v}_{up}\right\}$ | ▹ pointwise scaling of the distribution ${\mathbf{u}}^{*}$ | |

13: $\mathbf{v}\leftarrow \mathbf{v}+({F}_{r,1},\dots ,{F}_{r,m})$ | ▹ pointwise addition of functions | |

14: ${x}_{r}\leftarrow {x}_{r}+1$ | ▹$\mathbf{x}=({x}_{1},\dots ,{x}_{n})$ | |

15: ${v}_{*}\leftarrow $ the maximum of $\mathbf{v}$ | ||

16: $c\leftarrow $ the index of ${v}_{*}$ in $\mathbf{v}$ | ||

17: ${v}_{low}\leftarrow $ the minimum of $\left\{{v}_{*}/k,{v}_{low}\right\}$ | ▹ pointwise scaling of the distribution ${v}_{*}$ | |

18: $\mathbf{u}\leftarrow \mathbf{u}+({F}_{1,c},\dots ,{F}_{n,c})$ | ▹ pointwise addition of functions | |

19: ${y}_{c}\leftarrow {y}_{c}+1$ | ▹ $\mathbf{y}=({y}_{1},\dots ,{y}_{m})$ | |

20: exit the loop upon convergence | ▹ concrete condition given below | |

21: end for | ||

22: Normalize $\mathbf{x},\mathbf{y}$ to unit total sum | ▹ turn $\mathbf{x},\mathbf{y}$ into probability distributions. | |

23: return ${\mathbf{p}}^{*}\leftarrow \mathbf{x},{\mathbf{q}}^{*}\leftarrow \mathbf{y},$ and $F({\mathbf{p}}^{*},{\mathbf{q}}^{*})\leftarrow {\sum}_{i,j}{F}_{ij}\left(r\right)\xb7{x}_{i}\xb7{y}_{j}$ | ▹ $\approx {\left({\mathbf{p}}^{*}\right)}^{\top}\mathbf{A}{\mathbf{q}}^{*}$ |

## Notes

1 | i.e., a set such that $\varnothing \notin \mathcal{U},\mathbb{N}\in \mathcal{U}$, closed under ⊇-relation, and closed under finite intersections. |

2 | The mixed case is discussed in [23], but was found to be practically not meaningful, since it would be difficult to interpret the semantics of a comparison of categories to real numbers that are not ranks. |

3 | Inductively, we can define $a{\le}_{\mathrm{lex}}b:\u27faa\le b$ for $a,b\in \mathbb{R}$, and for $({a}_{n},\dots ,{a}_{1}),({b}_{n},\dots ,{b}_{1})$ we put $\mathbf{a}{\le}_{\mathrm{lex}}\mathbf{b}:\u27fa[({a}_{n}\le {b}_{n})\wedge ({a}_{n-1},\dots ,{a}_{1}){\le}_{\mathrm{lex}}({b}_{n-1},\dots ,{b}_{1})]$ |

4 | Instantly visible by considering any null-sequence $0<{x}_{n}\to 0$ as $n\to \infty $, satisfying $({x}_{n},0){>}_{\mathrm{lex}}(0,1)$, while ${lim}_{n\to \infty}({x}_{n},0)=(0,0){\le}_{\mathrm{lex}}(0,0)$. |

5 | For zero-sum games, two arbitrary equilibria $({\mathbf{x}}^{*},{\mathbf{y}}^{*})$ and $({\tilde{\mathbf{x}}}^{*},{\tilde{\mathbf{y}}}^{*})$ give rise to two more equilibria $({\mathbf{x}}^{*},{\tilde{\mathbf{y}}}^{*})$ and $({\tilde{\mathbf{x}}}^{*},{\mathbf{y}}^{*})$, which may not be the case for non-zero-sum games. A proof of convexity of the set for zero-sum games is found in [39]. |

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**Figure 1.**Distribution-valued game as a version of (14) with uncertainty around the expected payoffs (on the abscissa of each probability density).

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**MDPI and ACS Style**

Rass, S.; König, S.; Schauer, S.
Games over Probability Distributions Revisited: New Equilibrium Models and Refinements. *Games* **2022**, *13*, 80.
https://doi.org/10.3390/g13060080

**AMA Style**

Rass S, König S, Schauer S.
Games over Probability Distributions Revisited: New Equilibrium Models and Refinements. *Games*. 2022; 13(6):80.
https://doi.org/10.3390/g13060080

**Chicago/Turabian Style**

Rass, Stefan, Sandra König, and Stefan Schauer.
2022. "Games over Probability Distributions Revisited: New Equilibrium Models and Refinements" *Games* 13, no. 6: 80.
https://doi.org/10.3390/g13060080